#GWF2020 7-9 April 2020 / / / Amsterdam www.geospatialworldforum.org CLICK TO KNOW MORE
Prof Is mat E lhas s an is is mat@ks mat@ks u.edu.s u.edu.s a a K S U, R iyadh, S audi Arabia 1
• Introduction • Problem Definition • Importance of Space Resection • Importance of Space Resection • Photogrammetry versus Computer Vision • Mathematical Indirect Solutions • Mathematical Indirect Solutions • Direct Solutions • Comparison • Comparison • Conclusions 2
Photogrammetry is defined as a measurement technique Photogrammetry is defined as a measurement technique where the coordinates of the points in 3D of an object are calculated after measurements made on 2D photographic images taken by metric camera. photographic images taken by metric camera. The position and attitude of the camera (camera exterior The position and attitude of the camera (camera exterior orientation elements) during the exposure is an important factor in determining the required ground coordinates. The process of determining the position and attitude of the camera is called Space Resection (SR). the camera is called Space Resection (SR). 3
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The diagram below shows the SR problem The diagram below shows the SR problem
Importance: The problem is important for both photogrammetry The problem is important for both photogrammetry and computer vision disciplines. Some of the photogrammetric applications of the space resection are: space resection are: - Fixing ground coordinates by intersection from single photo after solving the space resection problem. problem. - Photo Triangulation, using multiple photos (Bundle Adjustment) - Camera calibration (Tsai, 1987) - Head-Mounted Tracking System Positioning (Azuma and Ward, 1991) and Ward, 1991) - Object Recognition - Ortho photo rectification 6
Importance of S Importance of S R R in C in C omputer Vision omputer Vision SR applications in computer vision include:: SR applications in computer vision include:: - Head-Mounted Tracking System Positioning (Azuma and Ward, 1991) (Azuma and Ward, 1991) - Robot picking and Robot navigation - Robot picking and Robot navigation (Linnainmaa et al. 1988) - Visual surveying in 3D input devices - Head pose computation - Head pose computation 7
S S pace R pace R esection: esection: P hotogrammetry-C omputer Vision Space resection is therefore dealt with in both Space resection is therefore dealt with in both photogrammetry & computer vision. Here is a comparison between the two treatments. Here is a comparison between the two treatments. Difference is shown in terminology and in processing: Terms: Terms: Photogrammetry V Computer Vision Camera exterior orientation Camera pose estimation parameters 8
P P hotogrammetry hotogrammetry V V C C omputer Vision omputer Vision T erms, coordinates Space resection Space resection Perspective n-point Perspective n-point problem problem Homogenous 3D Cartesian 3D Cartesian Coordinates Coordinates Spatial coordinates Camera line elements Of camera persp center Camera attitiude Camera attitiude Camera pose angles Camera pose angles 9
P hotogrammetry V C omputer Vision S S olution C olution C oncepts oncepts Non-linear problem Non-linear problem Linear problem Direct Solution Iterative Solution Initial Approximations Initial Approximations No need for initial No need for initial needed approximations Collinearity Condition Collinearity Condition Closed Form 10
Approximate S Approximate S olutions olutions Here are some approximate solutions of the SR problem: problem: 1 - The Direct Linear Transformation (DLT), which is a method frequently used in photogrammetry and remote method frequently used in photogrammetry and remote sensing. 2- The Church method proposed as a solution for single image resection (Slama, 1 image resection (Slama, 1 980). 980). 3- A simplified absolute orientation method based on object-distances and vertical lines used when no control object-distances and vertical lines used when no control points are available. This method is largely applied in archaeology and architecture by non-photogrammetrists due to its simplicity. due to its simplicity. 11
Approximate S Approximate S olutions olutions Continued: Continued: 4- A method of 3D conformal coordinate transformations (Dewitt, 1 transformations (Dewitt, 1 996) where a special 996) where a special formulation of the rotation matrix as a function of the azimuth and tilt is proposed. the azimuth and tilt is proposed. 5- An approximate solution of the spatial transformation (Kraus, 1 transformation (Kraus, 1 997) which is particularly 997) which is particularly suitable when incomplete control points are used. 12
Church’ solution (1 945) First iterative solution was published by Church, First iterative solution was published by Church, 1945, 1948. This is done by linearizing collinearity equations, This is done by linearizing collinearity equations, (Wolf, 1980; Salama, 1980) (Wolf, 1980; Salama, 1980) It needs a good starting value which constitutes an approximate solution. approximate solution. Approximate initial values which can be known to 10% 10% accuracy for scale and distances and to accuracy for scale and distances and to 5 o for rotation angles would then be 5 o for rotation angles would then be within 1 within 1 adjusted by the solution. See Seedahmed, 2008 for autonomous initial See Seedahmed, 2008 for autonomous initial values for exterior orientation parameters (EOP) 13
L y Collinearity Condition Collinearity Condition Z f f x a x a a Tilted photo y a plane x O The exposure station The exposure station Y of a photograph (L), an object Z L point (A) and its photo point (A) and its photo A A Z a image (a) all lie along X a a straight line (L, a, A). X L X L Y a Y a Y L X 14
Image & Ground Coordinate Systems Image & Ground Coordinate Systems • • Ground Coordinate System - X, Y, Z Ground Coordinate System - X, Y, Z z’ y’ y’ L Z x’ In Ground Coordinate System z a’ Exposure Station Coordinates L( X L , Y L , Z L ) • y a’ X X a’ Object Point (A) Coordinates A( X a , Y a , Z a ) • • Image coordinate system (x’, y’, z’ ) parallel to ground coordinate system (XYZ) ground coordinate system (XYZ) • • Z L Z L Y A In image Coordinate System Z a image point (a) coordinates a(x a ’, y a ’, z a ’) X a x a ’ , y a ’ and z a ’ are related to the measured Y a photo coordinates xa, ya, focal length (f) and the X L three rotation angles omega, phi and kappa. Y L Y L X 15
Developed in a sequence of three independent two-dimensional rotations. • rotation about x’ axis x 1 = x’ 1 y 1 = y’Cos + z’Sin z 1 = -y’Sin + z’Cos f rotation about y’ axis f rotation about y’ axis • • x 2 = -z 1 Sinf + x 1 Cosf y 2 = y 1 z 2 = z 1 Cosf + x 1 Sinf z 2 = z 1 Cosf + x 1 Sinf • rotation about z’ axis x = x 2 Cos + y 2 Sin y = -x 2 Sin + y 2 Cos y = -x 2 Sin + y 2 Cos z = z 2 16
x = x’(Cos f Cos ) + y’(Sin Sin f Cos + Cos Sin ) + z’( - Cos Sin f Cos + Sin Sin ) y = x’( - Cos f Sin ) + y’( - Sin Sin f Sin + Cos Cos ) + z’(Cos Sin f Sin + Sin Cos ) z = x’(Sin f ) + y’(- Sin Cos f ) + z’(Cos Cos f ) X = MX’ x = m 11 x’ + m 12 y’ + m 13 z’ y = m 21 x’ + m 22 y’ + m 23 z’ Rotation Matrix x m m m x ' 11 12 13 z = m x’ + m y’ + m z’ z = m 31 x’ + m 32 y’ + m 33 z’ y m m m y ' 21 22 23 z m m m z ' 31 32 33 The sum of the squares of the three “direction cosines” in any row or in any The sum of the squares of the three “direction cosines” in any row or in any column is unity. M - 1 = M T X’ = M T X 17
Collinearity Condition Equations Collinearity Condition Equations from Similar Triangles Collinearity condition equations developed from similar triangles (Wolf) x ' y ' z ' a a a x = m 11 x’ + m 12 y’ + m 13 z’ X X X X Y Y Y Y Z Z Z Z y = m 21 x’ + m 22 y’ + m 23 z’ y = m 21 x’ + m 22 y’ + m 23 z’ A A L L A A L L L L A A z = m 31 x’ + m 32 y’ + m 33 z’ X X Y Y Z Z A L A L A L x m z ' m z ' m z ' a 11 a 12 a 13 a Z Z Z Z Z Z * Dividing x a and y a by z a * Dividing x a and y a by z a A L A L A L X X Y Y Z Z * Substitute – f for z a A L A L A L y m z ' m z ' m z ' a 21 a 22 a 23 a Z Z Z Z Z Z A L A L A L * Correcting the offset of Principal X X Y Y Z Z A L A L A L point (x o , y o ) point (x o , y o ) z z m m z z ' ' m m z z ' ' m m z z ' ' a a 31 31 a a 32 32 a a 33 33 a a Z Z Z Z Z Z A L A L A L m ( X X ) m ( Y Y ) m ( Z Z ) 11 A L 12 A L 13 A L x x f a o m ( X X ) m ( Y Y ) m ( Z Z ) 31 32 33 A L A L A L m m ( ( X X X X ) ) m m ( ( Y Y Y Y ) ) m m ( ( Z Z Z Z ) ) 21 21 A A L L 22 22 A A L L 23 23 A A L L y y f a o ( ) ( ) ( ) m X X m Y Y m Z Z 31 A L 32 A L 33 A L 18
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